What is the intuition behind variational inference for Bayesian neural networks?
Your description of what is going on is more or less correct, although I am not completely sure that you have really understood it, given your last question.
So, let me enumerate the steps.

The computation of the posterior is often intractable (given that the evidence, i.e. the denominator of the right-hand side of the Bayes' rule, might be numerically expensive to approximate/compute or there's no closed-form solution)

To address this intractability, you cast the Bayesian inference problem (i.e. the application of Bayes' rule) as an optimization problem

You assume that you can approximate the posterior with another simpler distribution (e.g. a Gaussian), known as the variational distribution

You formulate this optimization problem as the minimization of some notion of distance (e.g. the KL divergence) between the posterior and the VD

However, the KL divergence between the posterior and the VD turns out to be intractable too, given that, if you expand it, you will find out that there's still an evidence term

Therefore, you use a tractable surrogate (i.e. equivalent, up to some constant) objective function, which is known as the evidence lower bound (ELBO) (which is sometimes known as the variational free energy), which is the sum of 2 terms

KL divergence between the VD and the prior
the likelihood of the parameters given the data





To address your last doubt/question, the ELBO does not contain the posterior (i.e. what you really want to find), but only the variational distribution (you choose this!), the prior (which you also define/choose), and the likelihood (which, in practice, corresponds to the typical usage of the cross-entropy; so the only thing that you need more, with respect to the traditional neural networks, is the computation of the KL divergence): in other words, you originally formulate the problem as the minimization of the KL divergence between the posterior and the VD, but this is just a formulation.